Exam 16: Additional Topics in Time Series Regression

"Heteroskedasticity typically occurs in cross-sections, while serial correlation is typically observed in time-series data." Discuss and critically evaluate this statement.

A VAR with five variables, 4 lags and constant terms for each equation will have a total of

A multiperiod regression forecast h periods into the future based on an AR(p)is computed

Volatility clustering

Some macroeconomic theories suggest that there is a short-run relationship between the inflation rate and the unemployment rate. How would you go about forecasting these two variables? Suggest various alternatives and discuss their advantages and disadvantages.

If X_t and Y_t are cointegrated, then the OLS estimator of the coefficient in the cointegrating regression is

You have re-estimated the two variable VAR model of the change in the inflation rate and the unemployment rate presented in your textbook using the sample period 1982:I (first quarter)to 2009:IV. To see if the conclusions regarding Granger causality of changed, you conduct an F-test for this new sample period. The results are as follows: The F-statistic testing the null hypothesis that the coefficients on Unemp_t-1, Unemp_t-2, Unemp_t-3, and Unempl_t-4 are zero in the inflation equation (Equation 16.5 in your textbook)is 6.04. The F-statistic testing the hypothesis that the coefficients on the four lags of ΔInf_t are zero in the unemployment equation (Equation 16.6 in your textbook)is 0.80. a. What is the critical value of the F-statistic in both cases? b. Do you think that the unemployment rate Granger-causes changes in the inflation rate? c. Do you think that the change in the inflation rate Granger-causes the unemployment rate?

The coefficients of the VAR are estimated by

Multiperiod forecasting with multiple predictors

Under the VAR assumptions, the OLS estimators are

The following is not an appropriate way to tell whether two variables are cointegrated:

Your textbook so far considered variables for cointegration that are integrated of the same order. For example, the log of consumption and personal disposable income might both be I(1)variables, and the error correction term would be I(0), if consumption and personal disposable income were cointegrated. (a)Do you think that it makes sense to test for cointegration between two variables if they are integrated of different orders? Explain. (b)Would your answer change if you have three variables, two of which are I(1)while the third is I(0)? Can you think of an example in this case?

A VAR with k time series variables consists of

Assume that you have used the OLS estimator in the cointegrating regression and test the residual for a unit root using an ADF test. The resulting ADF test statistic has a

To test the null hypothesis of a unit root, the ADF test

The BIC for the VAR is

In this case, the Granger causality statistic does not exceed the critical value, and hence the conclusion is that the change in the inflation rate does not Granger-cause the unemployment rate. $\widehat{\ln \mathrm { f }}$ _t = 0.05 - 0.31 ΔInf_{t-1 (0.14)(0.07)} t = 1982:I - 2009:IV, R² = 0.10, SER = 2.4 a. Calculate the one-quarter-ahead forecast of both ΔInf_2010:I and Inf_2010:I (the inflation rate in 2009:IV was 2.6 percent, and the change in the inflation rate for that quarter was -1.04). b. Calculate the forecast for 2010:II using the iterated multiperiod AR forecast both for the change in the inflation rate and the inflation rate. c. What alternative method could you have used to forecast two quarters ahead? Write down the equation for the two-period ahead forecast, using parameters instead of numerical coefficients, which you would have used.

ARCH and GARCH models are estimated using the

You have collected quarterly data for real GDP (Y)for the United States for the period 1962:I (first quarter)to 2009:IV. a. Testing the log of GDP for stationarity, you run the following regression (where the lag length was determined using the AIC): $\widehat{\Delta \ln Y_{t}}=0.03-0.0024 \ln Y_{t-1}+0.253 \Delta \ln Y_{t-1}+0.167 \Delta \ln Y_{t-2}$ $(0.03)\quad(0.0014)\quad\quad(0.072)\quad\quad(0.072)$ $t=1962: \mathrm{I}-2009: \mathrm{IV}, R^{2}=0.16, \text { SER }=0.008$ Use the ADF statistic with an intercept only to test for stationarity. What is your decision? b.You have decided to test the growth rate of real GDP for stationarity for the same sample period. The regression is as follows:S ${\widehat{\Delta^{2} \ln Y_{t}}}=0.0041-0.543 \Delta \ln Y_{\mathrm{t}-1}-0.186 \Delta 2 \ln Y_{\mathrm{t}-1}$ $(0.0009)(0.082)\quad\quad(0.071)$ $t=1962: \mathrm{I}-2009: \mathrm{IV}, R^{2}=0.16, S E R=0.008$ Use the ADF statistic with an intercept only to test for stationarity. What is your decision?c. integration terminology, what order of integration is the log level of real GDP? The growth rate? d. Given that the SER hardly changed in the second equation, why is the regression R²larger?

Think of at least five examples from economics where theory suggests that the variables involved are cointegrated. For one of these cases, explain how you would test for cointegration between the variables involved and how you could use this information to improve forecasting.